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CBSE Class 12 Mathematics Solved Question Paper 2010

Short Answer Type

If A = $[\begin{matrix} cosα & - sinα \\ sinα & cosα \end{matrix}]$ , then for what value of α is A an identity matrix?

What is the principal value of $\cos^{- 1} (- \frac{\sqrt{3}}{2})$ ?

What is the range of the function f(x) = $\frac{|x - 1|}{(x - 1)}$ ?

Find the minor of the element of second row and third column ( a₂₃ ) in the following determinant:

$|\begin{matrix} 2 & - 3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & - 7 \end{matrix}|$

If $[\begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix}] [\begin{matrix} 3 & 1 \\ 2 & 5 \end{matrix}] = [\begin{matrix} 7 & 11 \\ k & 23 \end{matrix}]$ , then write the value of k.

Evaluate: $\int \frac{\log x}{x} dx$

7.
What is the cosine of the angle which the vector $\sqrt{2} \hat{i} + \hat{j} + k$ makes with y-axis?

The y-axis can be represented in vector form by $\hat{j} and - \hat{j} .$

$Let \vec{a} = \sqrt{2} \hat{i} + \hat{j} + k and \vec{b} = \hat{j} or - \hat{j} cosθ = \frac{\hat{a} . b}{|\hat{a}| |b|} ∴ cosθ = \frac{(\sqrt{2} \hat{i} + \hat{j} + k) . (\pm \hat{j})}{|\sqrt{2} \hat{i} + \hat{j} + k| . |\hat{j}|} = \pm \frac{1 x 1}{\sqrt{{(\sqrt{2})}^{2} + {(1)}^{2} + {(1)}^{2}} x \sqrt{1^{2}}} = \pm \frac{1}{\sqrt{4}} = \pm \frac{1}{2} So, the cosine of the angle which the vector \sqrt{2} \hat{i} + \hat{j} + k makes with y - axis is \pm \frac{1}{2}$